New alternative statistic for testing several independent samples of correlation matrices in high dimension data
Jennrich Jennrich statistic is one of the existing statistics which is used for testing the equality of several independent samples of correlation matrices. The statistic is gaining considerable importance in several areas of economics and financial markets. In these research areas, the number of v...
Saved in:
Main Author: | |
---|---|
Format: | Thesis |
Language: | eng eng eng |
Published: |
2018
|
Subjects: | |
Online Access: | https://etd.uum.edu.my/6889/1/DepositPermission_s95832.pdf https://etd.uum.edu.my/6889/2/s95832_01.pdf https://etd.uum.edu.my/6889/3/s95832_02.pdf |
Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
id |
my-uum-etd.6889 |
---|---|
record_format |
uketd_dc |
institution |
Universiti Utara Malaysia |
collection |
UUM ETD |
language |
eng eng eng |
advisor |
Sharif, Shamshuritawati |
topic |
QA273-280 Probabilities Mathematical statistics |
spellingShingle |
QA273-280 Probabilities Mathematical statistics Atiany, Tareq A.M. New alternative statistic for testing several independent samples of correlation matrices in high dimension data |
description |
Jennrich Jennrich statistic is one of the existing statistics which is used for testing the equality of several independent samples of correlation matrices. The statistic is
gaining considerable importance in several areas of economics and financial markets. In these research areas, the number of variables, p, is usually larger than the sample size, n, which is known as high dimension data p > n. Subsequently, the estimation of correlation and covariance determinant will breakdown due to singularity problem. When this happens, Jennrich statistic is unable to function as the calculation involves the inversion of correlation matrix. Therefore, to resolve the aforementioned problem, this study develops an alternative statistic for testing several independent samples of correlation matrices in high dimension data. For this reason, the algebraic approach on the basis of vec operator, commutation matrix and Frobenius norm of
upper-off-diagonal elements are used to derive the new asymptotic distribution for the new alternative statistic, namely * Z statistic. Simulation study was conducted by considering different number of variables, sample sizes, and correlation shifts to evaluate the performance of the new statistic. In addition, real data on Asia Pacific currencies structure during the Tohoku earthquake were applied to validate the new * Z statistic. The power of the * Z statistic is compared with the existing Jennrich
statistic, and * T statistic through simulation study. As a result, the power of * Z statistic dominates the power of Jennrich statistic and * T statistic in all conditions, especially, when the shift in correlation matrix is at least 0.3 As a conclusion, the theoretical and simulation results are established and supported by desirable power of test. Meanwhile, investigation on real data indicates that the new alternative statistic can accommodate high dimension data. |
format |
Thesis |
qualification_name |
Ph.D. |
qualification_level |
Doctorate |
author |
Atiany, Tareq A.M. |
author_facet |
Atiany, Tareq A.M. |
author_sort |
Atiany, Tareq A.M. |
title |
New alternative statistic for testing several independent samples of correlation matrices in high dimension data |
title_short |
New alternative statistic for testing several independent samples of correlation matrices in high dimension data |
title_full |
New alternative statistic for testing several independent samples of correlation matrices in high dimension data |
title_fullStr |
New alternative statistic for testing several independent samples of correlation matrices in high dimension data |
title_full_unstemmed |
New alternative statistic for testing several independent samples of correlation matrices in high dimension data |
title_sort |
new alternative statistic for testing several independent samples of correlation matrices in high dimension data |
granting_institution |
Universiti Utara Malaysia |
granting_department |
Awang Had Salleh Graduate School of Arts & Sciences |
publishDate |
2018 |
url |
https://etd.uum.edu.my/6889/1/DepositPermission_s95832.pdf https://etd.uum.edu.my/6889/2/s95832_01.pdf https://etd.uum.edu.my/6889/3/s95832_02.pdf |
_version_ |
1747828123562409984 |
spelling |
my-uum-etd.68892021-08-09T04:38:44Z New alternative statistic for testing several independent samples of correlation matrices in high dimension data 2018 Atiany, Tareq A.M. Sharif, Shamshuritawati Awang Had Salleh Graduate School of Arts & Sciences Awang Had Salleh Graduate School of Arts and Sciences QA273-280 Probabilities. Mathematical statistics Jennrich Jennrich statistic is one of the existing statistics which is used for testing the equality of several independent samples of correlation matrices. The statistic is gaining considerable importance in several areas of economics and financial markets. In these research areas, the number of variables, p, is usually larger than the sample size, n, which is known as high dimension data p > n. Subsequently, the estimation of correlation and covariance determinant will breakdown due to singularity problem. When this happens, Jennrich statistic is unable to function as the calculation involves the inversion of correlation matrix. Therefore, to resolve the aforementioned problem, this study develops an alternative statistic for testing several independent samples of correlation matrices in high dimension data. For this reason, the algebraic approach on the basis of vec operator, commutation matrix and Frobenius norm of upper-off-diagonal elements are used to derive the new asymptotic distribution for the new alternative statistic, namely * Z statistic. Simulation study was conducted by considering different number of variables, sample sizes, and correlation shifts to evaluate the performance of the new statistic. In addition, real data on Asia Pacific currencies structure during the Tohoku earthquake were applied to validate the new * Z statistic. The power of the * Z statistic is compared with the existing Jennrich statistic, and * T statistic through simulation study. As a result, the power of * Z statistic dominates the power of Jennrich statistic and * T statistic in all conditions, especially, when the shift in correlation matrix is at least 0.3 As a conclusion, the theoretical and simulation results are established and supported by desirable power of test. Meanwhile, investigation on real data indicates that the new alternative statistic can accommodate high dimension data. 2018 Thesis https://etd.uum.edu.my/6889/ https://etd.uum.edu.my/6889/1/DepositPermission_s95832.pdf text eng public https://etd.uum.edu.my/6889/2/s95832_01.pdf text eng public https://etd.uum.edu.my/6889/3/s95832_02.pdf text eng public Ph.D. doctoral Universiti Utara Malaysia Aitkin, M., Nelson, W., & Reinfurt, K. H. (1968). Tests for correlation matrices. Biometrika, 327-334. Aktas, B. K. A. (2013). STATISTICAL POWER ANALYSIS. The 7th International Days of Statistics and Economics (pp. 578-587). Prague. Alfaro, J., & Ortega, J. F. (2009). A comparison of robust alternatives to Hotelling’s T 2 control chart. Journal of Applied Statistics, 36(12), 1385-1396. Anderson. (2006). Distance-based tests for homogeneity of multivariate dispersions. Biometrics, 62(1), 245-253. doi: 10.1111/j.1541-0420.2005.00440.x Anderson, T. W. (2003). Introduction to Multivariate Statistical Analysis. New York: John Wiley & Sons, Inc. Annaert, J., Claes, A. G., & De Ceuster, M. J. (2006). Inter-temporal Stability of the European Credit Spread Co-movement Structure. The European Journal of Finance, 12(1), 23-32. Annaert, J., De Ceuster, M. J., & Claes, A. G. (2003). Inter-temporal Stability of the European Credit Spread Co-movement Structure. University of Antwerp: Faculty of Applied Economics. Aslam, S., & Rocke, D. M. (2005). A robust testing procedure for the equality of covariance matrices. Computational statistics & data analysis, 49(3), 863- 874. Atiany, T. A. M., & Sharif, S. (2015). Asia-Pacific Currencies Structure Aftermath Tohoku Earthquake. Research Journal of Applied Sciences, Engineering and Technology, 11(2), 215-220. Atiany, T. A. M., & Sharif, S. (2016). New Statistical Test for Testing Several Correlation Matrices. Global Journal of Pure and Applied Mathematics, 12(5), 4285-4298. Bai, J., & Shi, S. (2011). Estimating high dimensional covariance matrices and its applications. 164 Barnett, I., & Onnela, J.-P. (2016). Change point detection in correlation networks. Scientific reports, 6. Baroudi, J. J., & Orlikowski, W. J. (1989). The problem of statistical power in MIS research. MIS Quarterly, 87-106. Bartlett, M., & Rajalakshman, D. (1953). Goodness of fit tests for simultaneous autoregressive series. Journal of the Royal Statistical Society. Series B (Methodological), 107-124. Bickel, P. J., & Levina, E. (2004). Some theory for Fisher's linear discriminant function,'naive Bayes', and some alternatives when there are many more variables than observations. Bernoulli, 989-1010. Bickel, P. J., & Levina, E. (2008a). Covariance regularization by thresholding. The Annals of Statistics, 2577-2604. Bickel, P. J., & Levina, E. (2008b). Regularized estimation of large covariance matrices. The Annals of Statistics, 199-227. Botman, D. P., de Carvalho Filho, I., & Lam, W. R. (2013). The curious case of the yen as a safe haven currency: a forensic analysis. Working Paper 13/228. Box, G. E. (1949). A general distribution theory for a class of likelihood criteria. Biometrika, 36(3/4), 317-346. Browne, M., & Shapiro, A. (1986). The asymptotic covariance matrix of sample correlation coefficients under general conditions. Linear Algebra and its Applications, 82, 169-176. Cai, T. T., Zhang, C.-H., & Zhou, H. H. (2010). Optimal rates of convergence for covariance matrix estimation. The Annals of Statistics, 38(4), 2118-2144. Cai, T. T., & Zhou, H. H. (2012). Minimax estimation of large covariance matrices under ℓ 1-norm. Statistica Sinica, 12(4), 1319-1349. Chen, X., Wang, Z. J., & McKeown, M. J. (2012). Shrinkage-to-tapering estimation of large covariance matrices. Signal Processing, IEEE Transactions on, 60(11), 5640-5656. Chesnay, F., & Jondeau, E. (2001). Does correlation between stock returns really increase during turbulent periods? Economic Notes, 30(1), 53-80. 165 Cho, D. C., & Taylor, W. M. (1987). The seasonal stability of the factor structure of stock returns. The Journal of Finance, 42(5), 1195-1211. Choi, Y.-G., Lim, J., Roy, A., & Park, J. (2016). Positive-definite modification of covariance matrix estimators via linear shrinkage. arXiv preprint arXiv:1606.03814. Chou, Y.-M., Mason, R. L., & Young, J. C. (2001). The control chart for individual observations from a multivariate non-normal distribution. Communications in statistics-Theory and methods, 30(8-9), 1937-1949. Clark‐Carter, D. (1997). The account taken of statistical power in research published in the British Journal of Psychology. British Journal of Psychology, 88(1), 71-83. Cohen, J. (1977). Statistical power analysis for the behavioral sciences (revised ed.): New York: Academic Press. Cohen, J. (1990). Things I have learned (so far). American psychologist, 45(12), 1304. Cooper, W. H., Donnelly, J. M., & Johnson, R. (2011). Japan’s 2011 earthquake and tsunami: economic effects and implications for the United States. Congressional research service. Cui, X. (2010). Computing the Nearest Correlation Matrix using Difference Map Algorithm. University of Waterloo, Canada. Da Costa Jr, N., Nunes, S., Ceretta, P., & Da Silva, S. (2005). Stockmarket comovements revisited. Economics Bulletin, 7(3). Da Rocha, G. N. V. (2008). Sparsity and Model Selection Through Convex Penalties: Structured Selection, Covariance Selection and Some Theory. University of California, Berkeley. Deblauwe, F., & Le, H. (2000). Stability of correlation between credit and market risk over different holding periods. Ph. D. Thesis, University of Antwerp Management School, Antwerp, Belgium. Djauhari, M. A. (2007). A measure of multivariate data concentration. Journal of Applied Probability and Statistics, 2(2), 139-155. 166 Djauhari, M. A., & Gan, S. L. (2014). Dynamics of correlation structure in stock market. Entropy, 16(1), 455-470. Djauhari, M. A., & Herdiani, E. (2008). Monitoring the Stability of Correlation Structure in Drive Rib Production Process: An MSPC Approach. Open Industrial & Manufacturing Engineering Journal, 1, 8-18. Eichholtz, P. M. (1996). Does international diversification work better for real estate than for stocks and bonds? Financial analysts journal, 52(1), 56-62. El Karoui, N. (2007). On spectral properties of large dimensional correlation matrices and covariance matrices computed from elliptically distributed data. University of California, Berkeley: Technical report from Department of Statistics. Fama, E. F. (1965). The behavior of stock-market prices. The journal of Business, 38(1), 34-105. Furrer, R., & Bengtsson, T. (2007). Estimation of high-dimensional prior and posterior covariance matrices in Kalman filter variants. Journal of Multivariate Analysis, 98(2), 227-255. Gali, S. (2015, 3-5 April). On Importance of Normality Assumption in Using a TTest: OneSample and Two Sample Cases. Paper presented at the Proceedings of the International Symposium on Emerging Trends in Social Science Research Chennai-India. Gan, S. L., Djauhari, M. A., & Ismail, Z. (2014). Monitoring correlation structures stability in foreign exchange market. Paper presented at the 2014 IEEE International Conference on Industrial Engineering and Engineering Management. Gande, A., & Parsley, D. C. (2005). News spillovers in the sovereign debt market. Journal of Financial Economics, 75(3), 691-734. Gilbert, N., & Troitzsch, K. (2005). Simulation for the social scientist: McGraw-Hill Education (UK). Goetzmann, W. N., Li, L., & Rouwenhorst, K. G. (2005). Long-term global market correlations. Journal of Business, 78(1). 167 Górski, A., Drozdz, S., & Kwapien, J. (2008). Minimal spanning tree graphs and power like scaling in FOREX networks. Acta Physica Polonica A, 114(3), 531-538. Haddad, F. S. F. (2013). Statistical Process Control Using Modified Robust Hotelling's T² Control Charts. Universiti Utara Malaysia. Herdiani, E. T. (2008). Statistik Penguji Kestabilan Barisan Matriks Korelasi. Institut Teknologi Bandung. Herdiani, E. T., & Djauhari, M. A. (2012). Distribution Sampling of Vector Variance without Duplications. World Academy of Science, Engineering and Technology, International Journal of Mathematical, Computational, Physical, Electrical and Computer Engineering, 6(12), 1673-1676. Hood, M., Kamesaka, A., Nofsinger, J., & Tamura, T. (2013). Investor response to a natural disaster: Evidence from Japan's 2011 earthquake. Pacific-Basin Finance Journal, 25, 240-252. Hotelling, H. (1940). The selection of variates for use in prediction with some comments on the general problem of nuisance parameters. The Annals of Mathematical Statistics, 11(3), 271-283. Imamura, F., & Anawat, S. (2011). Damage due to the 2011 Tohoku earthquake tsunami and its lessons for future mitigation. Paper presented at the Proceedings of the international symposium on engineering lessons learned from the 2011 Great East Japan Earthquake, March 1-4, 2012,, Tokyo, Japan. Jang, W., Lee, J., & Chang, W. (2011). Currency crises and the evolution of foreign exchange market: Evidence from minimum spanning tree. Physica A: Statistical Mechanics and its Applications, 390(4), 707-718. Jennrich, R. I. (1970). An asymptotic χ2 test for the equality of two correlation matrices. Journal of the American Statistical Association, 65(330), 904-912. Kaplanis, E. C. (1988). Stability and forecasting of the comovement measures of international stock market returns. Journal of international Money and Finance, 7(1), 63-75. Karoui, N. E. (2008). Operator norm consistent estimation of large-dimensional sparse covariance matrices. The Annals of Statistics, 2717-2756. 168 Khare, K., Oh, S. Y., & Rajaratnam, B. (2015). A convex pseudolikelihood framework for high dimensional partial correlation estimation with convergence guarantees. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(4), 803-825. Kollo, T., & Ruul, K. (2003). Approximations to the distribution of the sample correlation matrix. Journal of multivariate analysis, 85(2), 318-334. Kollo, T., & Von Rosen, D. (2006). Advanced multivariate statistics with matrices (Vol. 579): Springer Science & Business Media. Kullback, S. (1967). On testing correlation matrices. Applied Statistics, 80-85. Larntz, K., & Perlman, M. D. (1985). A simple test for the equality of correlation matrices. Rapport technique, Department of Statistics, University of Washington, 141. Lawley, D. (1963). On testing a set of correlation coefficients for equality. The Annals of Mathematical Statistics, 34(1), 149-151. Lee, S. (1998). The inter-temporal stability of real estate returns: an empirical investigation. Paper presented at the International Real Estate Conference Maastricht The Netherlands. Magnus, J. R., & Neudecker, H. (1980). The elimination matrix: some lemmas and applications. SIAM Journal on Algebraic Discrete Methods, 1(4), 422-449. Makridakis, S. G., & Wheelwright, S. C. (1974). An analysis of the interrelationships among the major world stock exchanges. Journal of Business Finance & Accounting, 1(2), 195-215. Maldonado, R., & Saunders, A. (1981). International portfolio diversification and the inter-temporal stability of international stock market relationships, 1957-78. Financial Management, 54-63. Mantegna, R. N., & Stanley, H. E. (2000). An introduction to econophysics: correlation and complexity in finance. Cambridge, UK: Cambridge University. Mason, R. L., Chou, Y.-M., & Young, J. C. (2009). Monitoring variation in a multivariate process when the dimension is large relative to the sample size. Communications in Statistics—Theory and Methods, 38(6), 939-951. 169 McCrum-Gardner, E. (2010). Sample size and power calculations made simple. International Journal of Therapy and Rehabilitation, 17(1), 10. Meric, I., & Meric, G. (1997). Co-movements of European equity markets before and after the 1987 crash. Multinational Finance Journal, 1(2), 137-152. Mizuno, T., Takayasu, H., & Takayasu, M. (2006). Correlation networks among currencies. Physica A: Statistical Mechanics and its Applications, 364, 336- 342. Montgomery, D. C. (2005). Introduction to statistical quality control. Fifth Edition: John Wiley & Sons. Mori, N., Takahashi, T., Yasuda, T., & Yanagisawa, H. (2011). Survey of 2011 Tohoku earthquake tsunami inundation and run‐up. Geophysical research letters, 38(7). Muirhead, R. J. (1982). Aspects of multivariate statistical analysis. JOHN WILEY & SONS, INC., 605 THIRD AVE., NEW YORK, NY 10158, USA, 1982, 656. Murphy, K. R., Myors, B., & Wolach, A. (2014). Statistical power analysis: A simple and general model for traditional and modern hypothesis tests: Routledge. Neudecker, H., & Wesselman, A. M. (1990). The asymptotic variance matrix of the sample correlation matrix. Linear Algebra and its Applications, 127, 589- 599. Nguyen, H. M. (2012). Natural disasters and the economies of Asian countries: The case of earthquakes and tsunamis in Japan, storms and floods in Vietnam. Vietnam National University, Hanoi: University of Economics and Business. Nickisch, S., Nockemann, C., Tillack, G.-R., Murphy, J., & Sturges, D. (1997). Alternative approaches to the validation of nondestructive testing methods Review of Progress in Quantitative Nondestructive Evaluation (pp. 2037- 2044): Springer. Olkin, I., Lou, Y., Stokes, L., & Cao, J. (2015). Analyses of wine-tasting data: A tutorial. Journal of Wine Economics, 10(1), 4-30. Philippatos, G. C., Christofi, A., & Christofi, P. (1983). The inter-temporal stability of international stock market relationships: Another view. Financial Management, 63-69. 170 Pourahmadi, M. (2013). High-dimensional covariance estimation: with highdimensional data: John Wiley & Sons. Ragea, V. (2003). Testing correlation stability during hectic financial markets. Financial Markets and Portfolio Management, 17(3), 289-308. Rahman, M., Pearson, L. M., & Heien, H. C. (2006). A modified anderson-darling test for uniformity. Bulletin of the Malaysian Mathematical Sciences Society, 29(1). Rencher, A. C. (2003). Methods of multivariate analysis (Vol. 492): John Wiley & Sons. Rothman, A. J., Levina, E., & Zhu, J. (2009). Generalized thresholding of large covariance matrices. Journal of the American Statistical Association, 104(485), 177-186. Sabharwal, A., & Potter, L. (2002). Wald statistic for model order selection in superposition models. IEEE transactions on signal processing, 50(4), 956- 965. Sang, Y., Dang, X., & Sang, H. (2016). Symmetric Gini Covariance and Correlation. arXiv preprint arXiv:1605.02332. Satake, K. (2013). Tohoku, Japan (2011 Earthquake and Tsunami) Encyclopedia of Natural Hazards (pp. 1015-1018): Springer. Sawyer, A. G. (1982). Statistical power and effect size in consumer research. NAAdvances in Consumer Research Volume 09. Schermelleh-Engel, K., Moosbrugger, H., & Müller, H. (2003). Evaluating the fit of structural equation models: Tests of significance and descriptive goodnessof- fit measures. Methods of psychological research online, 8(2), 23-74. Schott, J. R. (1996). Testing for the equality of several correlation matrices. Statistics & probability letters, 27(1), 85-89. Schott, J. R. (1997). Matrix Analysis for Statistics. New York: John Wiley & Sons. Schott, J. R. (2001). Some tests for the equality of covariance matrices. Journal of Statistical Planning and Inference, 94(1), 25-36. 171 Schott, J. R. (2003). Kronecker product permutation matrices and their application to moment matrices of the normal distribution. Journal of multivariate analysis, 87(1), 177-190. Schott, J. R. (2005). Testing for complete independence in high dimensions. Biometrika, 92(4), 951-956. Schott, J. R. (2007a). A test for the equality of covariance matrices when the dimension is large relative to the sample sizes. Computational statistics & data analysis, 51(12), 6535-6542. Schott, J. R. (2007b). Testing the equality of correlation matrices when sample correlation matrices are dependent. Journal of Statistical Planning and Inference, 137(6), 1992-1997. Schott, J. R. (2016). Matrix analysis for statistics: John Wiley & Sons. Sewell, M. (2011). Characterization of financial time series. RN, 11(01), 01. Sharif, S. (2013). A new statistic to the theory of correlation stability testing in financial market. Universiti Teknologi Malaysia, Faculty of Science. Sharif, S., & Djauhari, M. A. (2014). Asymptotic derivation of T* statistic. Paper presented at the International Conference on Quantitative Sciences and its Applications (ICOQSIA 2014): Proceedings of the 3rd International Conference on Quantitative Sciences and Its Applications. Sharif, S., Ismail, S., Omar, Z., & Theng, L. H. (2016). Validation of Global Financial Crisis on Bursa Malaysia Stocks Market Companies via Covariance Structure. American Journal of Applied Sciences, 13(11), 1091-1095. Sheppard, K. (2008). Forecasting Covariances using High-Frequency Data and Positive Semi-Definite Matrix Multiplicative Error Models Song, S. (2011). Dynamic Large Spatial Covariance Matrix Estimation in Application to Semiparametric Model Construction via Variable Clustering: the SCE approach. arXiv preprint arXiv:1106.3921. Steiger, J. H. (1980). Tests for comparing elements of a correlation matrix. Psychological bulletin, 87(2), 245-251. 172 Sul, J. H., Han, B., & Eskin, E. (2011). Increasing power of groupwise association test with likelihood ratio test. Journal of Computational Biology, 18(11), 1611-1624. Sullivan, G. M., & Feinn, R. (2012). Using effect size-or why the P value is not enough. Journal of graduate medical education, 4(3), 279-282. Tang, G. Y. (1995). Stability of international stock market relationships across month of the year and different holding intervals. The European Journal of Finance, 1(3), 207-218. Tang, G. Y. (1998). The intertemporal stability of the covariance and correlation matrices of Hong Kong stock returns. Applied financial economics, 8(4), 359-365. Tulic, M. (2010). Estimation of large covariance matrices using results on large deviations. na, Vienna University of Technology. Watson, J. (1980). THE STATIONARITY OF INTER‐COUNTRY CORRELATION COEFFICIENTS: A NOTE. Journal of Business Finance & Accounting, 7(2), 297-303. Wright, S. P. (1992). Adjusted p-values for simultaneous inference. Biometrics, 1005-1013. Wu, W. B., & Pourahmadi, M. (2003). Nonparametric estimation of large covariance matrices of longitudinal data. Biometrika, 90(4), 831-844. Xue, L., Ma, S., & Zou, H. (2012). Positive-definite ℓ1-penalized estimation of large covariance matrices. Journal of the American Statistical Association, 107(500), 1480-1491. Xue, L., & Zou, H. (2014). Rank-based tapering estimation of bandable correlation matrices. Statistica Sinica, 24, 83-100. Yahaya, S. S. S. (2005). Robust statistical procedure for testing the equality of central tendency parameter under skewed distributions. Unpublished Ph. D. thesis, Universiti Sains Malaysia. |